I agree with BoneheadBass and MatousAc. “... does not have a left ear” sounds fine to me. Reported this for possible addition.
After all, if he doesn’t have one, it isn’t “his”, I suppose. (I guess you could say, "he doesn't have his left ear", but this sounds slightly odd to me, as if he left it on the dresser when he went to school. You could probably get away with this, though, because the meaning is clear from context.)
But if he is missing his left ear, then it was "his" to begin with (at least rhetorically, even if the actual ear never existed). (Even here, though, I suppose you could also say, "missing a left ear", but it doesn't sound quite as natural to me. You could say, "missing an ear", but he would only otherwise have one left ear; thus, we're talking about one specific left ear: "his".)